The Spectrality Of Self-similar Measures And Spectral Structure Of A Kind Of Moran Spectral Measures On R

Let μ be a compactly supported Borel probability measure on Rd.If there exists a countable set Λ such that the exponential family E(Λ)={e2πi+<λ,x>:λ∈ Λ} is an orthonormal base for the space L2(μ),then the measure μ is called a spectral measure and the set Λ is called a spectrum for μ.With the development of spectral theory,related issues in spectral measures have become a hot topic of fractal geometry and harmonic analysis.This dissertation is divided into two parts:one is what kind of self-similar mea-sures are spectral;one is to study the spectral structure of a class of Moran spectral measures,that is,to construct all possible spectra up to translations.In the first Chapter,we introduce the research background,motivation and main results of this article;In Chapter 2,we give the basic knowledge and necessary tools.The core content of this article is form Chapters 3 to 5,which are briefly introduced as follows.In Chapter 3,we study the spectrality of the self-similar measures.So far,all known works have strict requirements on the digit set generated from the self-similar measure(the cardinality of the digit set≤4,the consecutive digit set or satisfying the compatible pair condition),and most of them are sufficient conditions.In order to overcome the restrictions on the digit set,we introduce the zero sets of sign function on the digit set,and then give a necessary condition for a self-similar measure to be a spectral one.This is a substantial progress made in researching Laba-Wang conjecture.Under certain conditions(including all known results),we prove that the compatible pair is a necessary and sufficient condition for the self-similar spectral measure.It is worth noting that the condition cannot be improved.In Chapter 4,we focus on a self-similar measure μt,D,which is generated by a ratio t-1 and a product-form digits set D.Using the close relationship between tiles and spectral measures,we give a complete characterization of the spectrality of the measure μt,D Before that,the measures,whose spectrality were completely characterized,are only the self-similar measures generated by the consecutive digit set(Bernoulli convolution)or the three-element digit set.In Chapter 5,we study the spectral structure of a kind of Moran spectral mea-sures.More precisely,the Moran measure is generated by a sequence of integers {pn}and digit sets {Dn} with three elements.We completely classify all maximal orthog-onal sets and give some sufficient conditions for a maximal orthogonal set being a spectrum.In Chapter 6,we give some questions for further research.